US20050114423A1  Binary coded decimal addition  Google Patents
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 US20050114423A1 US20050114423A1 US10/720,940 US72094003A US2005114423A1 US 20050114423 A1 US20050114423 A1 US 20050114423A1 US 72094003 A US72094003 A US 72094003A US 2005114423 A1 US2005114423 A1 US 2005114423A1
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 G—PHYSICS
 G06—COMPUTING; CALCULATING; COUNTING
 G06F—ELECTRIC DIGITAL DATA PROCESSING
 G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
 G06F7/38—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation
 G06F7/48—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using noncontactmaking devices, e.g. tube, solid state device; using unspecified devices
 G06F7/491—Computations with decimal numbers radix 12 or 20.
 G06F7/492—Computations with decimal numbers radix 12 or 20. using a binary weighted representation within each denomination
 G06F7/493—Computations with decimal numbers radix 12 or 20. using a binary weighted representation within each denomination the representation being the natural binary coded representation, i.e. 8421code
 G06F7/494—Adding; Subtracting

 G—PHYSICS
 G06—COMPUTING; CALCULATING; COUNTING
 G06F—ELECTRIC DIGITAL DATA PROCESSING
 G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
 G06F7/38—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation
 G06F7/48—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using noncontactmaking devices, e.g. tube, solid state device; using unspecified devices
 G06F7/50—Adding; Subtracting
 G06F7/505—Adding; Subtracting in bitparallel fashion, i.e. having a different digithandling circuit for each denomination
 G06F7/506—Adding; Subtracting in bitparallel fashion, i.e. having a different digithandling circuit for each denomination with simultaneous carry generation for, or propagation over, two or more stages
Abstract
The binary coded decimal (BCD) adder circuit adds two BCD encoded operands, with an input carry bit, and produces a BCD encoded sum. The adder has three stages. The first stage receives two BCD encoded operands as inputs, groups the inputs into contiguous blocks of 4bits each, computes an intermediate sum vector and carry vector without considering the input carry bit, and also computes propagation and generate functions for each 4bit group. The second stage is a carry look ahead circuit which computes all carries from the input carry, and the propagate and generate functions of the 4bit groups from the first stage. The third stage adjusts the intermediate sum vector with precorrection factors which depend upon the input carry and the carries generated from the second stage and the carry vectors from the first stage.
Description
 The present invention relates to a binary coded decimal (BCD) addition, including an adder circuit.
 In BCD arithmetic, at least four binary bits are needed to represent the numbers from 0 to 9. There are sixteen possible bit combinations using four binary bits, but only ten (i.e, 0 to 9) are valid BCD digits. Traditionally, BCD adder circuits adjust the binary sum after the addition has been completed if the value is more than nine (i.e., 1001_{2}). For example, whenever the unadjusted sum of two BCD digits produced a carry out (i.e., when the sum exceeds 1001_{2}), the sum is corrected by adding 0110_{2}.
 A known BCD adder, described in “Logic and Computer Design Fundamentals”, M Morris Mano and Charles R Kime, Pearson Educational Asia, second edition, 2001 is shown in
FIG. 1 . The BCD adder circuit 5 uses a fourbit binary adder 6 to add two BCD operands A(3) to A(0) and B(3) to B(0), and an input carry C_{in }to produce an intermediate sum Z(3) to Z(0). The adder circuit 6 also includes two levels of logic gates 7, 8, 9 to calculate the BCD carry output Cout from the intermediate sum. If there is a BCD carry, a correction factor of 0110_{2 }is added with the intermediate sum using a further binary adder 10.  The BCD adder circuit 5 shown in
FIG. 1 thus uses 10 levels of gates to perform a 4bit addition. For a 16bit adder circuit, the corresponding binary adder circuit requires 16 levels of gates. However the critical path is the generation of the carry Cout, and gate levels required to implement this is 22 for 16bit BCD addition.  Another BCD adder circuit is taught in U.S. Pat. No. 4,805,131 (Adlietta et al, assigned to Digital Equipment Corporation), issued on Feb. 14, 1989. The BCD adder of Adlietta et al has three stages. In the first stage, 0110_{2 }is added with the input BCD operands without considering the input carry. In the second stage the sum and carry vectors are added using a carry look ahead network to generate final carry and propagate vectors. In the third stage a correction circuit adds 1010_{2 }if needed.
 The logic terms in the calculation of a FINAL_CARRY vector increases as the number of bits of the BCD operands increases. Also, the carry vector calculated is propagated through every bit position. The bit terms of the FINAL_CARRY vector must propagate through every bit position. As the bit position increases from least significant bit position 0 through most significant bit position 15, the number of logical terms required to produce the terms of the FINAL_CARRY vector also increases. Therefore, more sophisticated logic gates are required to produce the more significant bit terms and keep the number of logic levels and associated delay to a minimum.
 For 16bit BCD addition the BCD adder circuit of Adlietta et al uses only 12 gate levels compared to 22 gate levels in the BCD adder circuit of
FIG. 1 . But the logic terms required for the implementation of the FINAL_CARRY of BCD adder of Adlietta et al is much higher. Also, when the number of bits of the BCD operands increase, the logic terms for calculating the FINAL_CARRY also increases, which makes the implementation complex for more than 16bits. There also is no reusability, meaning extending similar structures when the number of bits of the input BCD operands increases.  There remains a need for BCD addition circuits with lesser a number of gate levels as well as lesser number of logic terms without requiring any sophisticated logic gates. There is also a need to increase the reusability of logic gates in such adders.
 The binary coded decimal (BCD) adder circuit adds two BCD encoded operands, with an input carry bit, and produces a BCD encoded sum. The adder includes three stages. The first stage receives two BCD encoded operands as inputs, groups the inputs into contiguous blocks of 4bits each, computes an intermediate sum vector and carry vector without considering the input carry bit, and also computes propagation and generate functions for each 4bit group. The second stage is a carry look ahead circuit which computes all carries from the input carry, and the propagate and generate functions of the 4bit groups from the first stage. The third stage adjusts the intermediate sum vector with, precorrection factors which depend upon the input carry and the carries generated from the second stage and the carry vectors from the first stage.
 This invention reduces the time required to perform a BCD addition by reducing in the number of gate levels of the critical path for adding two BCD operands of 16bits each with an input carry to 12. Only 15 gate levels of delay arise for adding two 64bit BCD encoded operands with an input carry. The gate levels thus increase only by three, from twelve to fifteen, when the input BCD operands bit size increases from sixteen to sixtyfour.
 Since the BCD addition is carried out with 4bit groups, the same logic structure can be extended to provide a 16, 32 or 64bit BCD adder with extra levels of carry look ahead circuit, which increases the reusability of the logic.

FIG. 1 shows a known BCD adder circuit. 
FIG. 2 shows a 16bit BCD adder circuit embodying the invention. 
FIG. 3 shows detail of the circuit ofFIG. 2 when an addition of a first group of bits is performed. 
FIG. 4 shows a circuit for calculating first intermediate sum and carry vectors. 
FIG. 5 shows a circuit for calculating second intermediate sum and carry vectors. 
FIG. 6 shows a circuit for calculating third intermediate sum and carry vectors. 
FIG. 7 shows a carry look ahead circuit. 
FIG. 8 shows a circuit for performing a SUM(0) correction. 
FIG. 9 shows a circuit for performing a SUM(1) correction. 
FIG. 10 shows a circuit for performing a SUM(2) correction. 
FIG. 11 shows a circuit for performing a SUM(3) correction. 
FIG. 12 shows a stage 1 circuit for a 64bit BCD adder circuit. 
FIG. 13 shows a carry look ahead circuit for the 64bit BCD adder circuit. 
FIG. 14 shows a sum correction circuit for the 64bit BCD adder circuit.  Overview
 The BCD addition of two operands each of n bits with an input carry bit will be described with reference to
FIG. 2 . A BCD adder circuit 12 generally has three stages, and the number of bits of the operands is n=16. The addition is performed after grouping the nbits into contiguous (or successive) groups of 4bits (i.e., bits 03, bits 47, bits 811, bits 1215).  The first stage, represented by four circuits 13, 14, 15, 16, generates intermediate sum and carry vectors and propagate and generate functions. The second stage has a carry look ahead circuit 17 to generate carries from the input carry, and the propagate and generate functions from the first stage. The third stage includes four circuits 18, 19, 20, 21 that generate a final sum from the intermediate sum vectors based on precorrection factors which depend on the input carry and the carries generated from carry look ahead circuit 17 and first stage circuits 13, 14, 15, 16.
 First Stage
 Consider the 16bit BCD addition of two BCD encoded operands A(15) to A(0) and B(15) to B(0).
 The addition of first group of 4bits A(3) to A(0) and B(3) to B(0) by the circuit 13 is explained with reference to
FIGS. 3, 4 , 5 and 6. The circuits 14, 15, 16 for the other three groups of 4bit addition logic are of the same form.  As shown in
FIG. 3 , three intermediate sum vectors FS[3:0], SS[3:0], and TS[3:0], (where “F”=first, “S”=second and “T”=third) and carry vectors FC[4:0], SC[3:0], TC3 to TC0 are calculated by sequential generator circuits 22, 23, 24. The third sum and carry vectors TS[3:0] and TC3 to TC0 are provided to a respective sum correction circuit 18 of the third stage.  The circuit 22 computes the first intermediate sum vector FS(3) to FS(0) and carry vector FC(4) to FC(1) by HalfAdders (XOR and AND gates) as shown in
FIG. 4 . The logical operations performed are: 
 FS(3)=A(3) XOR B(3) (gate 37)
 FS(2)=A(2) XOR B(2) (gate 35)
 FS(1)=A(1) XOR B(1) (gate 33)
 FS(0)=A(0) XOR B(0) (gate 31)
 FC(4)=A(3) AND B(3) (gate 38)
 FC(3)=A(2) AND B(2) (gate 36)
 FC(2)=A(1) AND B(1) (gate 34)
 FC(1)=A(0) AND B(0) (gate 32)
 FC(0)=‘0’
 The second intermediate sum vector of the first stage SS(3) to SS(0) and carry vector SC(3) to SC(2) are calculated by the circuit 23, including HalfAdders, as shown in detail in
FIG. 5 . The logical operations performed are: 
 SS(3)=FS(3) XOR FC(3) (gate 43)
 SS(2)=FS(2) XOR FC(2) (gate 41)
 SS(1)=FS(1) XOR FC(1) (gate 39)
 SS(0)=FS(0) XOR FC(0)=FS(0); since FC(0)=‘0’
 SC(3)=FS(2) AND FC(2) (gate 42)
 SC(2)=FS(1) AND FC(1) (gate 40)
 SC(1)=FS(0) AND ‘0’=‘0’; since FC(0)=‘0’

 SC(0)=‘0’
 The third intermediate sum vectors of the first stage TS(3) to TS(0) and a carry TC3 are calculated by the circuit 24, including HalfAdders, as shown in detail in
FIG. 6 . The logical operations performed are: 
 TS(3)=SS(3) XOR SC(3) (gate 46)
 TS(2)=SS(2) XOR SC(2) (gate 44)
 TS(1)=SS(1) XOR SC(1)=SS(1); since SC(1)=‘0’
 TS(0)=SS(0) XOR SC(0)=FS(0); since SC(0)=‘0’ and SS(0)=FS(0)
 TC3=SS(2) AND SC(2) (gate 45)
 TC2=SS(1) AND SC(1)=‘0’; since SC(1)=‘0’
 TC1=SS(0) AND SC(0)=‘0’; since SC(0)=‘0’
 TC0=‘0’
 Comparing
FIGS. 46 , it can be seen that the design has been optimized such that HalfAdder circuits reduce to XOR gates or even to being straight through in some instances.  The propagate function P30 and the generate function G30 for the first group of 4bits are calculated from the intermediate sum and carry vectors, by the combination logic shown in
FIG. 3 represented by gates 25, 26, 27, 28, 29, 30. The logical operations performed are: 
 P30=TS(3) AND TS(0) (gate 30)
 G30=(SS(1) AND (SS(3) OR SC(3))) OR (SS(3) AND (SS(2) OR SC(2))) OR
 FC(4) (gates 25, 26, 27, 28, 29)
 Thus the functions P30 and G30, along with other 4bit group propagate and generate functions P74, P118, P1512, G74, G118 and G1512, are inputs to the second stage carry look ahead circuit 17.
 Second Stage
 The second stage carry look ahead circuit 17, calculates the carry values of all groups of 4bits from the propagate and generate values of the first stage and the input carry bit C0 as shown in
FIG. 7 . 
 C4=G30 OR (P30 AND C0) (gates 47 and 57)
 C8=G74 OR (P74 AND G30) OR (P74 AND P30 AND C0) (gates 48, 49 and/or gate 58)
 C12=G18 OR (P118 AND G74) OR (P118 AND P74 AND G30) OR (P118 AND P74 AND P30 AND C0) (gates 50, 51, 52 and 59)
 C16=G1512 OR (P1512 AND G118) OR (P1512 AND P118 AND G74) OR (P1512 AND P118 AND P74 AND G30) OR (P1512 AND P118 AND P74 AND P30 AND C0) (gates 53, 54, 55, 56 and 60)
 These carries are provided to the third stage.
 The carry C16 will be calculated only for a 16bit BCD addition. If 32bit or 64bit BCD addition is required, then propagate function P150 and generate function G150 will be produced instead of C16 to be provided as an input to the 64bit carry look ahead network, shown in
FIG. 13 . The propagate and generate functions P150 and G150 are calculated as: 
 P150=P1512 AND P118 AND P74 AND P30
 G150=G1512 OR (P1512 AND G118) OR (P1512 AND P118 AND G74) OR (P1512 AND P118 AND P74 AND G30)
 The values P30, P74, P118 and P1512 are propagate functions and G30, G74, G118 and G1512 are generate functions from the first stage circuits 13,14,15 and 16 of
FIG. 2 .  Third Stage
 The third stage circuits 18, 19, 20, 21 adjust the sum with precorrection factors which depend on the carries generated from the first stage circuits 13, 14, 15, 16, the second stage carry look ahead circuit 17 and, in one case, the input carry C0.
 In general, the sum correction for a 4bit block(say N down to N−3) adds first stage 4bit sum vector TS[N:N−3] and a precorrection factor. The precorrection factor is one of the values of (TC_{N}000)_{2 }or (TC_{N}001)_{2 }or (TC_{N}110)_{2 }or (TC_{N}111)_{2 }which will be decided based on the carry C_{N−3 }of previous 4bits (N−4 down to N−7) and the carry C_{N+1 }of current 4 bits(N down to N−3).
 For example, the general steps for sum correction block of bits 7 to 4 (where N=7) are:
 Step 1
 One of the values “TC_{7 }0 0 0”, “TC_{7 }0 0 1”, “TC_{7 }1 1 0”, “TC_{7 }1 1 1” will be selected by C8 (Carry for bits 7 to 4) and C4 (carry of previous 4 bits i.e. bits 3 to 0).

 if C8=0 and C4=0=>selection of TC_{7}000. (TC_{7 }is a single bit having value ‘1’ or ‘0’)
 if C8=0 and C4=1=>selection of TC_{7}001
 if C8=1 and C4=0=>selection of TC_{7}110
 if C8=1 and C4=1=>selection of TC_{7}111
 Step 2
 The above selected value will be added with first stage SUM vector TS[N:N−3].
 The first of the 4bit group sum correction circuits, namely circuit 18, will now be explained in detail.
 The inputs to the sum correction circuit 18 are the third intermediate sum vectors TS(3) to TS(0), the intermediate carry TC3 from the first stage the input carry C0 and the carry C4 from the second stage carry look ahead circuit 17.
 If the Precorrection Factor (PF) is taken as PF(3) to PF(0), and the intermediate sum is TS(3) to TS(0), then the Final Sum will be obtained from the following steps, which have four levels of calculations.
 In the first level, a First Correction Sum vector FCS and a First Correction Carry vector FCC are calculated from the PF vector (PF(3) to PF(0)) and the TS vector (TS(3) to TS(0)) as shown in Table 1.
TABLE 1 First_Correction_Sum First_Correction_Carry FCS(3) = TS(3) XOR PF(3) FCC(3) = TS(2) AND PF(2) FCS(2) = TS(2) XOR PF(2) FCC(2) = TS(1) AND PF(1) FCS(1) = TS(1) XOR PF(1) FCC(1) = TS(0) AND PF(0) FCS(0) = TS(0) XOR PF(0) FCC(0) = ‘0’  In the second level, a Second Correction Sum SCS and a Second Correction Carry SCC are calculated from first level outputs FCS(3) to FCS(0) and FCC(3) to FCC(3) as shown in Table 2.
TABLE 2 Second_Correction_Sum Second_Correction_Carry SCS(3) = FCS(3) XOR FCC(3) SCC(3) = FCS(2) AND FCC(2) SCS(2) = FCS(2) XOR FCC(2) SCC(2) = FCS(1) AND FCC(1) SCS(1) = FCS(1) XOR FCC(1) SCC(1) = FCS(0) AND FCC(0) SCS(0) = FCS(0) XOR FCC(0) SCC(0) = ‘0’  In the third level, a Third Correction Sum TCS and Third Correction Carry TCC are calculated from second level outputs SCS(3) to SCS(0) and SCC(3) to SCC(0) as shown in Table 3.
TABLE 3 Third_Correction_Sum Third_Correction_Carry TCS(3) = SCS(3) XOR SCC(3) TCC(3) = SCS(2) AND SCC(2) TCS(2) = SCS(2) XOR SCC(2) TCC(2) = SCS(1) AND SCC(1) TCS(1) = SCS(1) XOR SCC(1) TCC(1) = SCS(0) AND SCC(0) TCS(0) = SCS(0) XOR SCC(0) TCC(0) = ‘0’  In the fourth level, a final SUM vector is calculated from the third level outputs TCS(3) to TCS(0) and TCC(3) to TCC(3) as shown in Table 4.
TABLE 4 Final_Sum SUM(3) = TCS(3) XOR TCC(3) SUM(2) = TCS(2) XOR TCC(2) SUM(1) = TCS(1) XOR TCC(1) SUM(0) = TCS(0) XOR TCC(0)  As mentioned, the Precorrection Factor PF(3) to PF(0) is, in part, dependent upon the carries C0 and C4. The Precorrection Factor PF(3) to PF(0) is given below for each of four cases:
If the C4=‘0’ and C0=‘0’, the PF(3) to PF(0) is (TC3 0 0 0) Case1
If the C4=‘0’ and C0=‘1’, the PF(3) to PF(0) is (TC3 0 0 1) Case2
If the C4=‘1’ and C0=‘0’, the PF(3) to PF(0) is (TC3 1 1 0) Case3
If the C4=‘1’ and C0=‘1’, the PF(3) to PF(0) is (TC3 1 1 1) Case4  Case 1
 The final SUM vector is obtained after substituting PF(3) to PF(0) values (TC3 0 0 1) in the First_Sum_Correction and First_Carry_Correction equations and then reducing the Second_Sum_Correction, Second_Carry_Correction, Third_Sum_Correction and Third_Carry_Correction equations. The finally reduced SUM vector is given below:
SUM(3)=TS(3) XOR TC3 equ03
SUM(2)=TS(2) equ02
SUM(1)=TS(1) equ01
SUM(0)=TS(0) equ00  Case 2
 The final SUM vector is obtained after substituting PF(3) to PF(0) values (TC3 0 0 1) in the First_Sum_Correction and First_Carry_Correction equations and then reducing the Second_Sum_Correction, Second_Carry_Correction, Third_Sum_Correction and Third_Carry_Correction equations. The finally reduced SUM vector is given below:
SUM(3)=TS(3) XOR TC3 XOR (TS(0) AND TS(1) AND TS(2)) equ13
SUM(2)=(TS(0) AND TS(1)) XOR TS(2) equ12
SUM(1)=TS(0) XOR TS(1) equ11
SUM(0)=NOT TS(0) equ10  Case 3
 The final SUM vector is obtained after substituting PF(3) to PF(0) values (TC3 1 1 0) in the First_Sum_Correction and First_Carry_Correction equations and then reducing the Second_Sum_Correction, Second_Carry_Correction, Third_Sum_Correction and Third_Carry_Correction equations. The finally reduced SUM vector is given below:
SUM(3)=TS(3) XOR TC3 XOR (TS(1) OR TS(2)) equ23
SUM(2)=TS(1) XOR (NOT TS(2)) equ22
SUM(1)=NOT TS(1) equ21
SUM(0)=TS(0) equ20  Case 4
 The final SUM vector is obtained after substituting PF(3) to PF(0) values (TC3 1 1 1) in the First_Sum_Correction and First_Carry_Correction equations and then reducing the Second_Sum_Correction, Second_Carry_Correction, Third_Sum_Correction and Third_Carry_Correction equations. The finally reduced SUM vector is given below
SUM(3)=TS(3) XOR TC(3) XOR (TS(0) OR TS(1) OR TS(2)) equ33
SUM(2)=(TS(0) OR TS(1)) XOR(NOT TS(2)) equ32
SUM(1)=TS(0) XOR (NOT TS(1)) equ31
SUM(0)=NOT TS(0) equ30  The implementation of SUM(0) correction circuit from equ00, equ10, equ20 and equ30 is shown in
FIG. 8 . The implementation of SUM(1) correction circuit from equ01, equ11, equ21 and equ31 is shown inFIG. 9 . The implementation of SUM(2) correction circuit from equ02, equ12, equ22 and equ32 is shown inFIG. 10 . The implementation of SUM(3) correction circuit from equ03, equ13, equ23 and equ33 is shown inFIG. 11 .  Clearly, there are equivalent circuits to those shown in
FIGS. 811 for each of the other sum correction circuits 19, 20, 21, to generate SUM[7:4], SUM[11:8], and SUM[15:12] values.  Final Sum
 The Final Sum of the BCD addition is the sequence of SUM[15:12], SUM[11:8], SUM[7:4], SUM [3:0]. A final carry, equal to C16, also results as an output.
 Critical Paths
 In the BCD addition circuit 12 as described, there are a number of critical paths in the three stages. Critical paths determine the number of logic levels required, and the speed at which addition can be performed.
 Critical paths in the first stage:
 CRITICAL_PATH_{—}1—The path through circuits 22, 23 and 24 which generates intermediate sum vector TS(3) to TS(0) from the BCD input operands A(3) to A(0) and B(3) to B(0) contains three XOR gates, and so the gate levels for this critical path is considered as 6.
 CRITICAL_PATH_{—}2a—The path through circuits 22, 23 and gates 26, 28, 30 from the BCD input operands to the generate function requires 7 gate levels.
 CRITICAL_PATH_{—}2b—The path through circuits 22, 23 and gates 25, 27, 30 from the BCD input operands to the generate function requires 7 gate levels.
 CRITICAL_PATH_{—}2c—The path through circuits 22, 23, 24 and gate 29 from the BCD input operands to the propagate function requires 7 gate levels.
 Critical paths in the second stage:
 CRITICAL_PATH_{—}3—The carry look ahead circuit for 16bit BCD addition (circuit 17 in
FIG. 3 ) requires two gate levels of logic.  Critical paths in the third stage:
 CRITICAL_PATH_{—}4—The SUM correction circuit's critical path (gates 77, 78 and MUX 81 of
FIG. 11 ) from first stage output intermediate sum vector to final SUM vector requires 6 levels of gates, of which 4 levels are calculated before receiving the carries from the carry look ahead circuit 17.  CRITICAL_PATH_{—}5—The path from the carry look ahead output carries to the final SUM which has 2 gate levels (MUX 62 or MUX 66 or MUX 73 or MUX 81).
 In the case of the 16bit BCD adder circuit, the critical path is through SUM correction circuit and the path is CRITICAL_PATH_{—}1 (6 gate levels)+CRITICAL_PATH_{—}4 (6 gate levels). So the 16bit BCD adder circuit has a delay of only 12 gate levels.
 In the case of a 64bit or 32 bits BCD adder circuit, the critical path is through the Carry look ahead circuit and the path is CRITICAL_PATH_{—}2a or CRITICAL_PATH_{—}2b or CRITICAL_PATH_{—}2c (all are 7 gate levels)+CRITICAL_PATH_{—}3 (6 gate levels)+CRITICAL_PATH_{—}5 (2 gate levels). Therefore, 64bit or 32bit BCD adder circuits have 15 gate levels of logic in the critical path.
 The 64Bit adder circuit
 The 16bit BCD adder of
FIG. 2 can be extended to a 32bit or 64bit form with extra carry look ahead network levels.  A 64bit BCD adder circuit includes three stages as discussed with reference to the 16bit adder 12 shown in
FIG. 2 . As shown inFIG. 12 , Stage 1 has sixteen logic circuits 90, 92, . . . , 118, 120 of the type identified by numeral 13 inFIG. 3 .  The circuits 90120 receive contiguous 4bit operands A[3:0] and B[3:0], A[7:4] and B[7:4], and so on, and generate respective intermediate sums TS[3:0], . . . TS[63:60], intermediate carries TC3, . . . , TC63, propagate functions P30, . . . , P6360, and generate functions G30, . . . , G6360.
 The second stage carry look ahead circuits are shown in
FIG. 13 . Five carry look ahead circuits 130, 132, 134, 136, 138 are required. These circuits are of the same for as the circuit 17 shown inFIG. 7 .  The carry look ahead circuits are in two substages: four circuits 130, 132, 134, 136 coupled with the first stage circuits 90120 to receive 4bit propagate and generate functions, and a single circuit 138, receiving 16bit propagate and generate functions from the first substage. The outputs from the second substage to the third stage are the carries C16, C32, C48, and C64.
 The third stage, shown in
FIG. 14 has sixteen sum correction circuits 150, 152, . . . , 178, 180. Each of these circuits is implemented in the same manner as, for example, sum correction circuit 18 described with reference toFIGS. 811 . Together, the sum correction circuits 150180 calculate SUM[3:0] to SUM [63:60] giving the sum result of the operands A[63:0]+B[63:0]+C_{0}. The final carry is the value C64.  The 64bit or 32bit BCD adder requires six gate levels as shown in
FIG. 13 (circuits 130, 132, 134, 136 and 138) in the second stage critical path.  The extension from 16bit to 64bit addition can be achieved with high reusability. A 64bit adder circuit can be built by extending (i.e. reusing) the components of a 16bit circuit four times, and adding only one extra level of a carry look ahead circuit.
 Conclusion
 The circuit described reduces the number of gate levels in the addition of two BCDencoded operands with an input carry and makes the BCD addition fast.
 While the invention contemplates equal length blocks for the operands, the embodiment taught is 4bit blocks. For the 16bit adder, the blocks are of contiguous bits from the least to most significant bits. The use of 4bit blocks is the most preferred, however, in that the fanin and fanout of carry look ahead logic is the least onerous to implement in hardware.
 Various alterations and modifications can be made to the techniques and arrangements described herein, as would be apparent to one skilled in the relevant art.
Claims (19)
1. A binary coded decimal adder circuit for adding two operands and an input carry to give a sum comprising:
a first stage receiving said operands, grouping said operands into equal length blocks of contiguous bits and, for each said block, logically computing from said operands an intermediate sum vector, an intermediate carry vector, a propagate function and a generate function;
a second stage receiving said input carry and receiving from each said block of said first stage the respective propagate function and generate function, and carry look ahead computing therefrom carries for each said block and an output carry; and
a third stage, having the same number of blocks as said first stage, each third stage block receiving a respective intermediate sum vector, and logically adjusting said intermediate sum vector by precorrection factors which depend upon a respective intermediate carry vector, at least two of the one of said second stage carries and said input carry; and
wherein outputs of said third stage logic blocks together represent said sum of operands.
2. The adder circuit of claim 1 , wherein said blocks are 4bits in length.
3. The adder circuit of claim 1 , wherein said operands are of 16bit length, and further wherein:
said first stage has a like logic circuit for each 4bit block;
said second stage has a single carry look ahead logic circuit; and
said third stage has a like sum correction logic circuit for each 4bit block.
4. The adder circuit of claim 1 , wherein said operands are of 64bit length, and further wherein:
said first stage has a like logic circuit for each 4bit block;
said second stage has a first carry look ahead level of four carry look ahead logic circuits providing partial carries to a second carry look ahead level of a single carry look ahead circuit; and
said third stage has a like sum correction circuit for each 4bit block.
5. The adder circuit of claim 1 , wherein said precorrection factors are calculated depending upon four cases of the logical combinations of two carries.
6. The adder circuit of claim 5 , wherein said precorrection factors include the carry from the intermediate carry vector and a bit pattern depending upon the relevant one of said four cases.
7. The adder circuit of claim 1 , wherein said intermediate sum vector and said intermediate carry vector are computed by three cascaded HalfAdder circuits.
8. The adder circuit of claim 1 , wherein said precorrection factors include one of four values decided on the basis of a carry from a preceding block and a carry from the current block.
9. A binary coded decimal adder circuit for adding two 4Nbit operands and an input carry to give a sum comprising:
a first stage receiving said operands, grouping said operands into N contiguous 4bit length blocks and, for each said block, logically computing from said operands an intermediate sum vector, an intermediate carry vector, a propagate function and a generate function;
a second stage receiving said input carry and receiving from each said block of said first stage the respective propagate function and generate function, and carry look ahead computing therefrom carries for each said block and an output carry; and
a third stage, having the same number of blocks as said first stage, each third stage block receiving a respective intermediate sum vector, and logically adjusting said intermediate sum vector by precorrection factors which depend upon a respective intermediate carry vector, at least two of the one of said second stage carries and said input carry; and
wherein outputs of said third stage logic blocks together represent said sum of operands.
10. The adder circuit of claim 9 , wherein said precorrection factors are calculated depending upon four cases of the logical combinations of two carries.
11. The adder circuit of claim 10 , wherein said precorrection factors include the carry from the intermediate carry vector and a bit pattern depending upon the relevant one of said four cases.
12. The adder circuit of claim 10 , wherein said precorrection factors are calculated from the carry C_{N−3 }of the previous 4bits and the carry C_{N+1 }of the current 4bits.
13. The adder circuit of claim 9 , wherein said intermediate sum vector and said intermediate carry vector are computed by three cascaded HalfAdder circuits.
14. A method for performing binary coded decimal addition of two operands and an input carry to give a sum comprising the steps of:
grouping said operands into equal length blocks of contiguous bits;
for each said block:
computing from said operands an intermediate sum vector, an intermediate carry vector, a propagate function and a generate function;
computing carries, including an output carry, from said input carry and the respective propagate function and generate function; and
logically adjusting the respective intermediate sum vector by precorrection factors which depend upon a respective intermediate carry vector, at least two of said carries and said input carry; and
combining said adjusted intermediate sum for all blocks to form said sum.
15. The method of claim 14 , wherein said precorrection factors are calculated depending upon four cases of the logical combinations of two carries.
16. The method of claim 15 , wherein said precorrection factors include the carry from the intermediate carry vector and a bit pattern depending upon the relevant one of said four cases.
17. The method of claim 14 , wherein said intermediate sum vector and said intermediate carry vector are computed by three cascaded HalfAdder circuits.
18. A computer program product, carried on a storage medium, for performing binary coded decimal addition of two operands and an input carry to give a sum including a computer program comprising:
code means for grouping said operands into equal length blocks of contiguous bits;
and for each said block:
code means for computing from said operands an intermediate sum vector, an intermediate carry vector, a propagate function and a generate function;
code means for computing carries, including an output carry, from said input carry and the respective propagate function and generate function; and
code means for logically adjusting the respective intermediate sum vector by precorrection factors which depend upon a respective intermediate carry vector, at least two of said carries and said input carry; and
code means for combining said adjusted intermediate sum for all blocks to form said sum.
19. A computer system for performing binary coded decimal addition of two operands and an input carry to give a sum comprising:
a storage unit for storing data and instructions to be performed by a processing unit; and
a processing unit coupled to said storage unit, said processing unit programmed to:
group said operands into equal length blocks of contiguous bits;
and for each said block to:
compute from said operands an intermediate sum vector, an intermediate carry vector, a propagate function and a generate function;
compute carries, including an output carry, from said input carry and the respective propagate function and generate function; and
logically adjust the respective intermediate sum vector by precorrection factors which depend upon a respective intermediate carry vector, at least two of said carries and said input carry; and
combine said adjusted intermediate sum for all blocks to form said sum.
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US7546328B2 (en) *  20040831  20090609  Wisconsin Alumni Research Foundation  Decimal floatingpoint adder 
US7743084B2 (en) *  20040923  20100622  Wisconsin Alumni Research Foundation  Processing unit having multioperand decimal addition 
EP2191329B1 (en) *  20070907  20160615  Precision Rubber Plate Co., Inc.  System and method for exposing a digital polymer plate 
US9143159B2 (en) *  20121004  20150922  Silminds, Inc.  DPD/BCD to BID converters 
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2003
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US3629565A (en) *  19700213  19711221  Ibm  Improved decimal adder for directly implementing bcd addition utilizing logic circuitry 
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US20080016140A1 (en)  20080117 
US7991819B2 (en)  20110802 
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